What is the integral of ((x^{1/3}+1)^{3/2})/(x^{2/3)} ?

The integral of ((x^{1/3}+1)^{3/2})/(x^{2/3)} is 6/5 (x^{1/3}+1)^{5/2}+C

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integral of ((x^{1/3}+1)^{3/2})/(x^{2/3)}

\int \frac{( x ^{\frac{1}{3}} + 1 ) ^{\frac{3}{2}}}{x ^{\frac{2}{3}}} d x

\frac{6}{5} (x^{\frac{1}{3}} + 1)^{\frac{5}{2}} + C

Solution steps

\int \frac{( x ^{\frac{1}{3}} + 1 ) ^{\frac{3}{2}}}{x ^{\frac{2}{3}}} d x

Apply u-substitution

= \int 3 (u + 1)^{\frac{3}{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int (u + 1)^{\frac{3}{2}} d u

Apply u-substitution

= 3 \cdot \int v^{\frac{3}{2}} d v

Apply the Power Rule

= 3 \cdot \frac{2}{5} v^{\frac{5}{2}}

Substitute back

= 3 \cdot \frac{2}{5} (x^{\frac{1}{3}} + 1)^{\frac{5}{2}}

Simplify

3 \cdot \frac{2}{5} (x^{\frac{1}{3}} + 1)^{\frac{5}{2}} : \frac{6}{5} (x^{\frac{1}{3}} + 1)^{\frac{5}{2}}

= \frac{6}{5} (x^{\frac{1}{3}} + 1)^{\frac{5}{2}}

Add a constant to the solution

= \frac{6}{5} (x^{\frac{1}{3}} + 1)^{\frac{5}{2}} + C

What is the integral of ((x^{1/3}+1)^{3/2})/(x^{2/3)} ?
The integral of ((x^{1/3}+1)^{3/2})/(x^{2/3)} is 6/5 (x^{1/3}+1)^{5/2}+C